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Data Abstraction

Programming Language Essentials

2nd edition

Chapter 2.2 An Abstraction for Inductive Data Types

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define-datatype bintreebintree: 'Number' | '(' 'Symbol' bintree bintree ')'

(define-datatype bintree bintree? (leaf-node (datum number?)) (interior-node (key symbol?) (left bintree?) (right bintree?)))

predicateconstructors

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Scheme 48 Imagehttp://www.cs.indiana.edu/eopl/code.zip

$ scheme48> ,open srfi-23> ,load r5rs.scm> ,load sllgen.scmsllgen.scm 2000-09-25 11:48> ,load define-datatype.scmdefine-datatype.scm version J3 2002-01-02> ,dump eopl.image EOPLWriting eopl.image

$ scheme48 -i eopl.image>

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Terminology

aggregate: contains values of other types, e.g., array or record with named fields.

union: values are of other types.

discriminated union: value contains explanatory tag and value of union's types.

variant record: discriminated union of record types.

Scheme values are discriminated union of primitive types.

Inductively defined data types can be represented as variant records.

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define-datatype

Scheme extension

(define-datatype type-name type-predicate-name

(variant-name

(field-name predicate) …

) …

)

creates variant record type with one or more variants with zero or more fields

type and variant names must be globally unique

must be used at top level

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define-datatype (2)

creates type-predicate-name predicate with one argument

creates variant-name constructor with one argument per field

(define-datatype bintree bintree?

(leaf-node

(datum number?))

(interior-node

(key symbol?)

(left bintree?)

(right bintree?)))

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define-datatype (3)(leaf-node 29)

(bintree? (leaf-node 29))

(interior-node 'foo (leaf-node 1) (leaf-node 2))

(bintree? (interior-node 'foo (leaf-node 1) (leaf-node 2)) )

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s-lists-list: '(' symbol-expression* ')'symbol-expression: 'Symbol' | s-list

(define-datatype s-list s-list? (empty-s-list) (non-empty-s-list (first symbol-exp?) (rest s-list?)))

(define-datatype symbol-exp symbol-exp? (symbol-symbol-exp (data symbol?)) (s-list-symbol-exp (data s-list?)))

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s-list based on Scheme lists(define-datatype s-list s-list?

(an-s-list

(data (list-of symbol-exp?))))

(define list-of ; returns predicate for list

(lambda (pred)

(lambda (val)

(or (null? val)

(and (pair? val)

(pred (car val))

((list-of pred) (cdr val))

) ) ) ) )

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Examples(s-list? (empty-s-list))

(s-list?

(non-empty-s-list (symbol-symbol-exp 'a)

(empty-s-list)

) )

(s-list? (an-s-list '()))

(s-list?

(an-s-list (list (symbol-symbol-exp 'a)))

)

(s-list?

(an-s-list (cons (symbol-symbol-exp 'a)

'()

) ) )

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Sum of leaves of bintree(define leaf-sum (lambda (tree) (cases bintree tree (leaf-node (datum) datum) (interior-node (key left right) (+ (leaf-sum left) (leaf-sum right))) ) ) )

(leaf-sum (interior-node 'foo (leaf-node 1) (leaf-node 2)) )

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cases(cases type-name expression

(variant-name (field-name …) body)

…

(else body)

)

value of expression must be of type-name

variant selects appropriate variant-name

each field value is bound to field-name and body is executed

without else, all variants must be specified

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Lambda Calculus Representationexpr: 'Symbol' | '(' 'lambda' '(' 'Symbol' ')' expr ')' | '(' expr expr ')'

(define-datatype expression expression? (var-exp (id symbol?)) (lambda-exp (id symbol?) (body expression?)) (app-exp (rator expression?) (rand expression?)))

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Examples(expression? (var-exp 'x))(expression? (lambda-exp 'x (var-exp 'x)))(expression? (app-exp (var-exp 'f) (var-exp 'x)))(expression? (lambda-exp 'x (app-exp (var-exp 'f) (var-exp 'x))) )

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Syntax and Representation

BNF specifies concrete syntax, external representation

define-datatype defines building blocks for abstract syntax, internal representation

expr: 'Symbol'

var-exp (id)

| '(' 'lambda' '(' 'Symbol' ')' expr ')'

lambda-exp (id body)

| '(' expr expr ')'

app-exp (rator rand)

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Abstract Syntax Tree lambda-exp

id body

x app-exp

rator rand

var-exp app-exp

id rator rand

f var-exp var-exp

id id

f x

(lambda (x) (f (f x)))

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Programming without an AST(define occurs-free?

(lambda (var exp)

(cond

((symbol? exp) (eqv? var exp))

((eqv? (car exp) 'lambda)

(and (not (eqv? (caadr exp) var))

(occurs-free? var (caddr exp))))

(else (or (occurs-free? var (car exp))

(occurs-free? var (cadr exp))

) ) ) ) )

plagued by c*r references

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Programming with an AST(define occurs-free? (lambda (var exp) (cases expression exp (var-exp (id) (eqv? id var)) (lambda-exp (id body) (and (not (eqv? id var)) (occurs-free? var body))) (app-exp (rator rand) (or (occurs-free? var rator) (occurs-free? var rand)) ) ) ) )

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Example(occurs-free?

'f

(lambda-exp 'x

(app-exp (var-exp 'f) (var-exp 'x))

) )

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unparse-expression(define unparse-expression (lambda (exp) (cases expression exp (var-exp (id) id) (lambda-exp (id body) (list 'lambda (list id) (unparse-expression body) ) ) (app-exp (rator rand) (list (unparse-expression rator) (unparse-expression rand)) ) ) ) )

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Example(unparse-expression

(lambda-exp 'x

(app-exp (var-exp 'f) (var-exp 'x))

) )

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parse-expression(define parse-expression

(lambda (datum)

(cond

((symbol? datum) (var-exp datum))

((pair? datum)

(if (eqv? (car datum) 'lambda)

(lambda-exp (caadr datum)

(parse-expression (caddr datum)))

(app-exp

(parse-expression (car datum))

(parse-expression (cadr datum))

) ) )

(else (error 'parse-expression datum))

) ) )

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Example(unparse-expression

(parse-expression

'(lambda (x) (f x))

)

)